14 Answers
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I have been meaning to ask this question for a while, but in a different form. This thread looks like a good place to share the motivation (and later I may ask the question I have in mind as a follow up).

I make my living applying mathematics and statistics to solving the kinds of problems a GIS is designed to address. One can learn to use a GIS effectively without knowing much math at all: millions of people have done it. But over the years I have read (and responded to) many thousands of questions about GIS and in many of these situations some basic mathematical knowledge, beyond what's usually taught (and remembered) in high school, would have been a distinct advantage.

The material that keeps coming up includes the following:

Trigonometry and spherical trigonometry. Let me surprise you: this stuff is overused. In many cases trig can be avoided altogether by using simpler, but slightly more advanced, techniques, especially basic vector arithmetic.

Elementary differential geometry. This is the investigation of smooth curves and surfaces. It was invented by C. F. Gauss in the early 1800's specifically to support wide-area land surveys, so its applicability to GIS is obvious. Studying the basics of this field prepares the mind well to understand geodesy, curvature, topographic shapes, and so on.

Topology. No, this does not mean what you think it means: the word is consistently abused in GIS. This field emerged in the early 1900's as a way to unify otherwise difficult concepts with which people had been grappling for centuries. These include concepts of infinity, of space, of nearness, of connectedness. Among the accomplishments of 20th century topology was the ability to describe spaces and calculate with them. These techniques have trickled down into GIS in the form of vector representations of lines, curves, and polygons, but that merely scratches the surface of what can be done and of the beautiful ideas lurking there. (For an accessible account of part of this history, read Imre Lakatos' Proofs and Refutations. This book is a series of dialogs within a hypothetical classroom that is pondering questions that we would recognize as characterizing the elements of a 3D GIS. It requires no math beyond grade school but eventually introduces the reader to homology theory.)

Differential geometry and topology also deal with "fields" of geometric objects, including the vector and tensor fields Waldo Tobler has been talking about for the latter part of his career. These describe extensive phenomena within space, such as temperatures, winds, and crustal movements.

Calculus. Many people in GIS are asked to optimize something: find the best route, find the best corridor, the best view, the best configuration of service areas, etc. Calculus underlies all thinking about optimizing functions that depend smoothly on their parameters. It also offers ways to think about and calculate lengths, areas, and volumes. You don't need to know much Calculus, but a little will go a long way.

Numerical analysis. We often have difficulties solving problems with the computer because we run into limits of precision and accuracy. This can cause our procedures to take a long time to execute (or be impossible to run) and can result in wrong answers. It helps to know the basic principles of this field so that you can understand where the pitfalls are and work around them.

Computer science. Specifically, some discrete mathematics and methods of optimization contained therein. This includes some basic graph theory, design of data structures, algorithms, and recursion, as well as a study of complexity theory.

Geometry. Of course. But not Euclidean geometry: a tiny bit of spherical geometry, naturally; but more important is the modern view (dating to Felix Klein in the late 1800's) of geometry as the study of groups of transformations of objects. This is the unifying concept to moving objects around on the earth or on the map, to congruence, to similarity.

Statistics. Not all GIS professionals need to know statistics, but it is becoming clear that a basic statistical way of thinking is essential. All our data are ultimately derived from measurements and heavily processed afterwards. The measurements and the processing introduce errors that can only be treated as random. We need to understand randomness, how to model it, how to control it when possible, and how to measure it and respond to it in any case. That does not mean studying t-tests, F-tests, etc; it means studying the foundations of statistics so that we can become effective problem solvers and decision makers in the face of chance. It also means learning some modern ideas of statistics, including exploratory data analysis and robust estimation as well as principles of constructing statistical models.

Please note that I am not advocating that all GIS practitioners need to learn all this stuff! Also, I am not suggesting that the different topics should be learned in isolation by taking separate courses. This is merely an (incomplete) compendium of some of the most powerful and beautiful ideas that many GIS people would deeply appreciate (and be able to apply) were they to know them. What I suspect we need is to learn enough about these subjects to know when they might be applicable, to know where to go for help, and to know how to learn more if it should be needed for a project or a job. From that perspective, taking a lot of courses would be overkill and would likely tax the patience of the most dedicated student. But for anyone who has an opportunity to learn some mathematics and has a choice of what to learn and how to learn it, this list might provide some guidance.

Excellent summary, the fundamentals of mathematics would go a long way in any GIS program.
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Dan PattersonFeb 25 '11 at 18:44

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(+1) It helps to have such an overview somewhere, especially for the naive user.
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chlFeb 26 '11 at 20:21

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Fortunately most GIS work does not require an understanding of quaternions! However, anyone who gets down to the vertex/coordinate level of processing vector data would benefit from knowing complex arithmetic: it reduces almost any relevant geometric operation to simple math.
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whuber♦Mar 6 '11 at 22:22

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+1 Excellent answer, and it emphasizes the main point well. Which is that this is more about preparing the mind for GIS concepts, as opposed to being something that you would use yourself. Often, students (myself included) will decry all of this work as useless, since "the computer does the math anyway," but that misses the point. You do end up using these mathematical ideas indirectly in GIS, even if you never look at another equation again (which is unlikely). It's about knowing your way around the underlying concepts.
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R ThiedeSep 29 '11 at 6:54

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I think that WHuber's list is fantastic. I would love to see the topics incorporated into courses - and not as a separate "math, calculus, statistic, trig course), but show how those wonderful concepts (not the scary terms) are useful for practical GIS problem solving. What a Course - I'm signing up.
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user10412Sep 23 '12 at 3:16

I think you are going to get a lot of differing opinions on this one. I had to take Calculus I and II (for a geology degree), and at the time, I suffered through them both. In hindsight, I really wish I would have taken more math courses. Not because I love math so much, but more because math really makes you think and learn how to solve problems in many different ways, and I see so, so many people who don't know how to think critically and solve problems, which in our line of work, is an invaluable skill.

So, my answer would be at least Calculus I, as that really puts everything you ever learned in algebra and trig to work for you, and it really makes you think.

The number of people who tell me they wish they had learned more math far outnumbers the very few who feel they learned too much! (All the latter are PhD mathematicians who ended up doing something else.)
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whuber♦Feb 25 '11 at 18:37

I have a pretty math heavy background and have never thought of it as a waste.

Geometry/Trig and algebra are a must.
Arguments can be made whether Calculus is or isn't necessary (three years may be excessive, but I would say at least one year is good).
Discrete Math is helpful for those who end up programming.

Others have touched quite a few things that I didn't even think of (statistics, analysis) that I didn't even think to lump in with my math. All of these are excellent. As many said, these aren't things that everyone needs to know, and it is very dependent on what you want to do with your career, but I would think that anyone who has a draw towards problem solving in the slightest should jump into quite a bit of math (see whuber's amazing list).
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MaryBethFeb 25 '11 at 19:37

Updated to address your MIT list--there is not one thing that I see on that list that couldn't prove useful. Of course this depends on what you get into. But many of those courses would apply. And, it's not just math, I know someone who got into GIS (primarily LiDAR, bathymetry, etc) with a Physics degree and he doesn't think it's a waste. ;)
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MaryBethFeb 25 '11 at 19:43

Agreed that three years of Calculus is heavily excessive for GIS, yes. I did one year of it though and I must say, it turned out to be essential. Not for everyday use, but for understanding the concepts involved.
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R ThiedeSep 29 '11 at 6:57

Geometry/Trig and Algebra as suggested by MaryBeth, would be a minimum, but this would be at the high school level (country-dependent, but normally grade 11 although 12 would be nice). This is particularly important in understandaing projections and transformations as well as as operations involving distance, direction and area calculations. Also, a course on algorithms (probably at the university level) would go a long way to understanding how some of the GIS functionality is carried out (eg. intersection, closest and the list goes on). For educators, the presumption of an appropriate math background should not be taken for granted (in my experience), you will/may have to provide the foundations yourself (gently) so as not to discourage those spatially-interested or inclined.

Core to GIS are Geometry, Trig and algebra. After this I would put calculus.

After that it depends on the area of GIS you want/decide to specialize in. I like application development more than analysis so the computer science side of things help me the most. On the other hand if you like the analysis/mapmatics side of things then statistics and modeling classes are the way to go (yeah SPSS - do they make this anymore?).

On a side note; GIS app development is becoming very language independent (agnostic?). A certain large GIS software developer is supporting APIs in many different flavors and a solid understanding of general programming is more valuable then an expertise in any particular one.

On the other hand when it come to GIS analysis the concepts are rooted firmly in fundamental mathematical disciplines. Algorithms using calc and stats seem to dominate (at lest from my limited view).

I'd hope for some exposure to linear algebra, computational geometry and statistics. Statistics I feel is especially important because it's the least 'dummy proof' area of functionality provided by commercial GIS software products.

Calculus can be a bit of a long road, but it is never a bad thing to know about differentiation and integration!

+1 for remembering computational geometry. As an analogy, it's a bit like having a professional driver study how the mechanical systems of a car work. Such a driver will use the vehicle efficiently and will know what to do in difficult situations or when things go wrong.
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whuber♦Dec 16 '11 at 18:56

Agree with dassouki, it really depends on what area you intend to focus on with GIS.

In Australia the biggest and most financially rewarding area is the mining industry. To become not just another GIS geek, if you were to understand Geology and Geophysics and the underlaying geophysical data, the world will be your oyster.

I hear often, that the lack of geological or geochemistry knowledge of the GIS pundents is a big issue. This is especially true when exploration geology is concerned. To understand the data you are using is very very crucial.

Physics is important for Oceanography GIS

Statistics very important in Urban and Regional Planning

Geometry for spacial awareness

Computer Science for programming GIS applications. Especially Python to be used as your computational mathematics.

Agreed. Though I'm relatively new to GIS, I have found a very diverse background in the field--engineers from every background, computer scientists, physicists, geologists, etc. It's one of the things I love, since there are so many different outlooks and ways of doing things, but it's a double edged sword as well, since there are so many different outlooks and different ways of doing things.
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MaryBethFeb 25 '11 at 20:00

I think this paper, "Energy-Information Transmission Tradeoff in Green Cloud Computing" offers a good example of the kinds of math future GIS Analysts should be exposed to. I don't think in-depth understanding of the theory is needed, just enough to know how to implement models based on either the methods described in the paper, or perhaps simplified methods. Imagine how much more interesting this paper would be if it were accompanied by a web based model. (maybe call it a data center geodesign tool)

Very interesting thought. The methods in this paper are discrete optimization. I agree that GIS analysts do not in general need to know the theory in detail, but it would be (actually, is) a valuable skill to be able to translate a real-world problem into this kind of model (which typically leads to some form of constrained nonlinear optimization), to select software that can solve it, to be able to apply reality checks to the output, and to interface it all with a GIS.
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whuber♦Mar 5 '11 at 22:23

Yes, I guess it's a depth vs. breadth issue. Maybe deciding what courses a prospective analyst needs is an optimization problem. Then later, there seems to be an tendency as one's career progresses to gain more and more expertise in a smaller and smaller subject area.
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Kirk KuykendallMar 6 '11 at 21:26

It also depends on which field you peruse. In my field, statistics and socio-economic type models (maximizing utility functions and such) seem to lead the way; however, other GIS-oriented fields require differing amounts of math.

It really all depends on what mess you get into; however, you don't need a huge understanding of math to get by, as long as you roughly understand the concepts, how to apply them and how to calculate the the equations, a thorough understanding of the subject isn't usually needed

As usual @whuber provide an insightful, through answer. I would add that the answer is dependent on the specific application of GIS your are interested in. This is a general term for a very large field of spatial applications. As such, course work should be guided by a specific focus of spatial analysis or computer science.

My particular focus is on spatial statistics in ecological applications. In this specific field of spatial analysis I guide students towards course work in matrix algebra and mathematical statistics. A background in probability theory, provided by mathematical statistics, can be quite helpful in understanding statistics in general and provide skills in the development of new methods. This requires a solid background in calculus and prerequisites of two semesters of upper-division calc are not uncommon.

Coursework in matrix algebra provide skills that aid in understanding the mechanisms behind spatial statistics and code-based (programming) implementation of complex spatial methods. Although I must add that I wholeheartedly agree with @whuber in that many complex spatial problems can be distilled into basic mathematical solutions.

Here is some coursework that I recommend for a mathematical background in spatial statistics that are available at the University of Wyoming. Obviously, I do not make my students take all of these courses, and associated prerequisites, but this is a good potential selection. ALthough, I do make all of my students take probability theory. Since your question was specific to mathematics I excluded coursework in statistics and quantitative ecology.

MATH 4500. Matrix Theory. The study of matrices, an important tool in statistics, physics, engineering and applied mathematics in general. Concentrates on the structure of matrices, including diagonalizability; symmetric, hermitian and unitary matrices; and canonical forms.

As GIS analyst with less than 6 months on the job, I can tell you that I wish I had studied more statistics. Intro to statistics + spatial statistics were a good start, but I find that there are a lot of problems with regression, probability, or data distributions that require reading material not covered in the 2 classes above. Getting experience with R, Matlab, or the like would have been invaluable. Machine Learning would also help.

I remember in university my professors would say it was vital that as a geologist my maths should be very good. Same thing was said by my GIS professor. Both were proved wrong as everything is automated nowadays =D

[cynical voice] As far as getting a job as a GIS Analyst is concerned, no mathematics is required, only 2-5 years of experience with the Esri family of GIS products. Don't believe me? Look at the job ads.
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martin fDec 24 '13 at 19:36